201110.6197v1 Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication文档.pdf

1 1 0 2 t c O 7 Some remarks on the two-variable main 2 ] conjecture of Iwasawa theory for elliptic curves T N without complex multiplication . h t Jeanine Van Order ∗ a m Abstract [ We establish several results towards the two-variable main conjecture 1 of Iwasawa theory for elliptic curves without complex multiplication over v imaginary quadratic fields, namely (i) the existence of an appropriate p- 7 adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic 9 structure theory of the dual Selmer group, following works of Coates, 1 Hachimori-Venjakob, et al., and (iii) the implications of dihedral or anti- 6. cyclotomic main conjectures with basechange. The result of (i) is deduced 0 from the construction of Hida and Perrin-Riou, which in particular is seen 1 to give a bounded distribution. The result of (ii) allows us to deduce a 1 corank formula for the p-primary part of the Tate-Shafarevich group of an 2 1 elliptic curve in the Z -extension of an imaginary quadratic field. Finally, 0 p (iii) allows us to deduce a criterion for one divisibility of the two-variable 2 main conjecture in terms of specializations to cyclotomic characters, fol- y lowing a suggestion of Greenberg, as well as a refinement via basechange. r o e